Math Problem Statement

f(x)= x−2 4x−5 ​ ,what is the equation of asympote

Solution

To find the equations of the asymptotes for the given rational function:

f(x)=x24x5f(x) = \frac{x - 2}{4x - 5}

Step 1: Vertical Asymptote

The vertical asymptotes occur when the denominator equals zero (and the numerator does not also equal zero at the same point). Set the denominator 4x5=04x - 5 = 0 and solve for xx:

4x5=0x=544x - 5 = 0 \quad \Rightarrow \quad x = \frac{5}{4}

So, the vertical asymptote is:

x=54x = \frac{5}{4}


Step 2: Horizontal Asymptote

For horizontal asymptotes, we compare the degrees of the numerator and denominator:

  • The degree of the numerator (x2x - 2) is 1.
  • The degree of the denominator (4x54x - 5) is 1.

When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficients are:

Numerator coefficient: 1Denominator coefficient: 4\text{Numerator coefficient: } 1 \quad \text{Denominator coefficient: } 4

Thus, the horizontal asymptote is:

y=14y = \frac{1}{4}


Final Answer

The equations of the asymptotes are:

  1. Vertical asymptote: x=54x = \frac{5}{4}
  2. Horizontal asymptote: y=14y = \frac{1}{4}

Would you like a detailed graph of this function or additional explanation?


Related Questions:

  1. How do you determine the oblique asymptote of a rational function if applicable?
  2. Can a function have more than one vertical asymptote?
  3. What happens to f(x)f(x) as x±x \to \pm \infty for this specific function?
  4. How do horizontal asymptotes differ from slant (oblique) asymptotes?
  5. Can the graph of f(x)f(x) intersect its horizontal asymptote?

Tip:

When identifying asymptotes, always check for cancellations in the rational function to ensure there are no "holes" that might affect the vertical asymptote.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Algebra

Formulas

Vertical Asymptote: Set denominator equal to zero and solve for x.
Horizontal Asymptote: If degrees of numerator and denominator are equal, asymptote is the ratio of leading coefficients.

Theorems

Vertical Asymptote Theorem
Horizontal Asymptote Theorem

Suitable Grade Level

Grades 9-11